OFFSET
0,4
COMMENTS
A002315(n) is a NSW number (named after Newman, Shanks, and Williams) and is an integer h that solves the Diophantine equation 2*n^2 = h^2 + 1.
Row n=0 is included by convention and corresponds to the Pythagorean triple 1^2 + 0^2 = 1^2.
The hypotenuses of all these Pythagorean triples are perfect squares.
REFERENCES
Miguel Ángel Pérez García-Ortega, José Manuel Sánchez Muñoz and José Miguel Blanco Casado, El Libro de las Ternas Pitagóricas, Preprint 2024.
FORMULA
EXAMPLE
Table begins:
n=0: 1, 0, 1;
n=1: 7, 24, 25;
n=2: 41, 840, 841;
n=3: 239, 28560, 28561;
n=4: 1393, 970224, 970225;
n=5: 8119, 32959080, 32959081;
MATHEMATICA
t[n_]:=t[n]= Module[{a, b}, a=((1+Sqrt[2])^(2n+1)-(Sqrt[2]-1)^(2n+1))/2; b=(a^2-1)/2; {a, b, b+1}]; ternas={}; Do[ternas=Join[ternas, FullSimplify[t[n]]], {n, 0, 10}]; ternas
CROSSREFS
KEYWORD
nonn,easy,tabf
AUTHOR
Miguel-Ángel Pérez García-Ortega, Oct 12 2024
STATUS
approved